These are the Cauchy–Riemann equations from complex analysis (see e.g. [1]})! These are fulfilled by analytic functions of either \(\mathcal {Z}^{\star }=x-{\rm i}y\) , in the case \(\lambda =\alpha \) , or \(\mathcal {Z}=x+{\rm i}y\) , when \(\lambda =-\alpha \) . Therefore the critical points of the Berry connection in a gyrotropic medium—which are, of course the reason the Chern number is non–zero—correspond to those points where the wave is an analytic function of position. At these points the wave depends solely on either the complex number \(\mathcal {Z}\) , or on \(\mathcal {Z}^{\star }\) , depending on the band of interest and the sign of the gyrotropy.
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